Chapter 3.3, Problem 55E

To determine

To calculate: The correct value of $y\text{'}\left(1\right)$.

Answer to Problem 55E

The correct option is (B).

Explanation of Solution

Given information:

The equation is $y=uv$.

Given values of the function

  $u\left(1\right)=2,u\text{'}\left(1\right)=3,v\left(1\right)=-1\text{and}v\text{'}\left(1\right)=1$

The given options are (A) $-4$ (B) $-1$ (C) $1$ (D) $4$ (E) $7$

Formula Used:

Product rule of differentiation:

  $\frac{d}{dx}uv=v\times \frac{d}{dx}u+u\times \frac{d}{dx}v$

Here $u,v$ are the functions.

Calculation:

The given expression is $y=uv$.

Differentiate the above function.

  $\begin{array}{c}\frac{d}{dx}y=\frac{d}{dx}uv\\ =v\times \frac{d}{dx}u+u\times \frac{d}{dx}v\\ =u\text{'}\times v+u\times v\text{'}\end{array}$

Now, substitute the values $u\left(1\right)=2,u\text{'}\left(1\right)=3,v\left(1\right)=-1\text{and}v\text{'}\left(1\right)=1$ in the above expression.

  $\begin{array}{c}\frac{d}{dx}y=3\times \left(-1\right)+2\times 1\\ =-3+2\\ =-1\end{array}$

Hence, the correct option is (B).